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The \(s\)-Hamiltonian index. (English) Zbl 1223.05166
Summary: For integers \(k,s\) with 0\(\leqslant k\leqslant s\leqslant |V(G)|-3\), a graph \(G\) is called \(s\)-Hamiltonian if the removal of any \(k\) vertices results in a Hamiltonian graph. For a simple connected graph that is not a path, a cycle or a \(K_{1,3}\) and an integer \(s\geqslant \)0, we define \(h_s(G)= \min \{m:L^m(G)\) is \(s\)-Hamiltonian \(\}\) and \(l(G)=\max\{m:G\) has a divalent path of length \(m\) that is not both of length 2 and in a \(K_{3}\}\), where a divalent path in \(G\) is a non-closed path in \(G\) whose internal vertices have degree 2 in \(G\). We prove that \(h_s(G)\leqslant l(G)+s+1\).

05C45 Eulerian and Hamiltonian graphs
Full Text: DOI
[1] Bondy, J.A.; Murty, U.S.R., Graph theory with applications, (1976), Macmillan London, Elsevier, New York · Zbl 1134.05001
[2] Chartrand, G.; Wall, C.E., On the Hamiltonian index of a graph, Studia sci. math. hungar., 8, 43-48, (1973) · Zbl 0279.05121
[3] Clark, L.H.; Wormald, N.C., Hamilton-like indices of graphs, Ars combin., 15, 131-148, (1983) · Zbl 0536.05046
[4] Harary, F.; Nash-Williams, C.St.J.A., On eulerian and Hamiltonian graphs and line graphs, Canad. math. bull., 8, 701-709, (1965) · Zbl 0136.44704
[5] Lai, H.-J., On the Hamiltonian index, Discrete math., 69, 43-53, (1988) · Zbl 0638.05034
[6] Mantel, W., Problem 28 (solution by H. gouwentak, W. mantel, J. texeira de mattes, F. schuh and W. A. Wythoff), Wiskundige opgaven, 10, 60-61, (1907)
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